Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative

Bartz, Sedi and Bauschke, Heinz H. and Borwein, Jonathan M. and Reich, Simeon and Wang, Xianfu (2007) Fitzpatrick functions, cyclic monotonicity and Rockafellar's antiderivative. Nonlinear Analysis Theory Methods and Applications, 66 (5). pp. 1198-1223.

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    Several deeper results on maximal monotone operators have recently found simpler proofs using Fitzpatrick functions. In this paper, we study a sequence of Fitzpatrick functions associated with a monotone operator. The first term of this sequence coincides with the original Fitzpatrick function, and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties. It is shown that for any maximal cyclically monotone operator, the pointwise supremum of the sequence of Fitzpatrick functions is closely related to Rockafellar's antiderivative. Several examples are explicitly computed for the purpose of illustration. In contrast to Rockafellar's result, a maximal 3-cyclically monotone operator need not be maximal monotone. A simplified proof of Asplund's observation that the rotation in the Euclidean plane by pi/n is n-cyclically monotone but not (n+1)-cyclically monotone is provided. The Fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail. We conclude with a new proof of Moreau's result concerning the convexity of the set of proximal mappings.

    Item Type: Article
    Uncontrolled Keywords: convex function, Fenchel conjugate, Fitzpatrick function, cyclically monotone operator, maximal monotone operator, subdifferential operator, sublinear function
    Subjects: UNSPECIFIED
    Faculty: UNSPECIFIED
    Depositing User: Dr David Allingham
    Date Deposited: 28 Sep 2012 12:05
    Last Modified: 11 Jan 2015 19:21

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